Scalable Bayesian Framework

• Scalable Bayesian Framework is an approach that ensures computational tractability in Bayesian inference by employing neural surrogates, partitioned data, and parallel processing techniques.
• It innovatively reduces traditional cubic or exponential complexities via surrogate modeling, Wasserstein barycenter aggregation, and product-form factorizations for efficient posterior approximation.
• Advanced methods such as coreset constructions, particle-based flows, and variational corrections are integrated to enhance performance in high-dimensional and deep Bayesian learning applications.

A scalable Bayesian framework is an approach to Bayesian inference or model learning that is explicitly designed to remain computationally tractable and accurate as data sizes, model complexity, or the required computational resources increase, often to the regime of large-scale or high-dimensional problems. These frameworks address central bottlenecks of traditional Bayesian inference—such as cubic complexity in sample size for Gaussian processes, exponential complexity in model selection, non-scalable MCMC algorithms, and challenges in massive parallel or distributed environments—by employing advanced algorithmic, mathematical, or systems-level innovations.

1. Surrogate Modeling and Linear-Scaling Bayesian Optimization

Traditional Bayesian optimization employs Gaussian process (GP) models that require inverting an $N \times N$ covariance matrix at each update, with $N$ the number of observations, resulting in $O(N^3)$ computational cost. As demonstrated in "Scalable Bayesian Optimization Using Deep Neural Networks" (Snoek et al., 2015), the use of a deep neural network as an adaptive basis function generator enables a fundamental scalability improvement. After training a deep network, the last hidden layer produces a $D$-dimensional feature vector $\phi(x)$. Bayesian linear regression is performed on these features, with predictive mean and variance:

$$\mu(x; \mathcal{D}, \Theta) = m^\top \phi(x) + \eta(x)$$

$$\sigma^2(x; \mathcal{D}, \Theta) = \phi(x)^\top K^{-1} \phi(x) + 1/\beta$$

where $K = \beta \Phi^\top \Phi + \alpha I$, and inversion is $O(D^3)$ (with $D\ll N$), while the update in $\Phi$ is $O(ND)$. This shift ensures linear scalability in $N$ and cubic complexity in the often small $D$. This framework enables massively parallel Bayesian optimization and large-scale hyperparameter search, as demonstrated on object recognition (CIFAR-10 and CIFAR-100) and image captioning (COCO), with competitive benchmarks achieved using thousands of parallel evaluations within days (Snoek et al., 2015).

2. Divide-and-Conquer, Posterior Aggregation, and Wasserstein Barycenters

Divide-and-conquer approaches partition massive data sets into smaller subsets, fit Bayesian models to each subset independently (e.g., via MCMC), and then aggregate via a principled composition of subset posteriors. "Scalable Bayes via Barycenter in Wasserstein Space" (Srivastava et al., 2015) introduced an approach where subset posteriors $\pi_m(\theta|Y_{[j]})$ are combined through their barycenter in the Wasserstein-2 ($W_2$) space:

$$\overline{\Pi}_n(\cdot|Y^{(n)}) = \arg\min_{\mu\in\mathcal{P}_2(\Theta)} \frac{1}{k}\sum_{j=1}^k W_2^2(\mu, \pi_m(\cdot|Y_{[j]}))$$

This geometric aggregation yields a posterior contraction rate close to the optimal parametric rate (up to logarithmic factors), with empirical results showing state-of-the-art approximations to the full posterior even for nonlinear functions of the parameters. The negligible cost of the barycenter computation (solved by linear programming or Sinkhorn algorithms) renders the method suited for massive-scale applications (as in MovieLens recommender systems) (Srivastava et al., 2015).

3. Product-Form Factorization and Parallelized Bayesian Graphical Models

For high-dimensional models such as Gaussian graphical models, scalable frameworks utilize factorized (product-form) pseudo-likelihoods. As in "A scalable quasi-Bayesian framework for Gaussian graphical models" (Atchade, 2015), the neighborhood selection method leads to a quasi-posterior that decomposes as a product over nodes:

$$\widehat{\mathcal{P}}_{n,p}(d\theta|x) = \prod_{j=1}^p \widehat{\mathcal{P}}_{n,p,j}(d\theta_{·j}|x, \sigma_j^2)$$

where each factor is a quasi-posterior for a regression subproblem. This approach allows MCMC sampling for each node's parameter vector entirely in parallel, enabling inference for models with $p>1000$. Theoretical results demonstrate contraction at rates determined by the hardest (most ill-posed) subproblem: $\sqrt{(s\log p)/n}$, with $s$ the maximum degree (Atchade, 2015). This design allows analysis of graph structures at scales unmanageable by traditional Bayesian methods for graphical models.

4. Block-Wise and Hierarchical Variable Selection

Scalable model selection is critical for high-dimensional variable selection and model averaging. "Scalable Bayesian variable selection and model averaging under block orthogonal design" (Papaspiliopoulos et al., 2016) partitions predictors into blocks (via block-diagonal approximations or spectral clustering). The posterior probability factorizes over blocks:

$$P(\gamma|y) \propto \prod_{b=1}^B L_b(\gamma_b)\pi(\gamma_b)$$

Resulting computational cost is $O(B2^m)$ (with $m$ the block size) versus $O(2^p)$. This enables deterministic or adaptive exploration of the model space, implemented efficiently in R (mombf package), and facilitates application to, e.g., gene expression and economic forecasting data (Papaspiliopoulos et al., 2016).

5. Coreset Construction for Automated Posterior Approximation

Coreset-based scalable Bayesian frameworks compress large data sets to weighted subsets, preserving posterior structure with greatly reduced computational cost. "Automated Scalable Bayesian Inference via Hilbert Coresets" (Campbell et al., 2017) reframes coreset construction in Hilbert space, choosing coresets that minimize inner-product (Hilbert-norm) error rather than worst-case sensitivity:

$$\|\mathcal{L}(w) - \mathcal{L}\|^2 = (w - 1)^\top K (w - 1)$$

where $K$ is the kernel of inner products of log-likelihood functions. Two algorithms—importance sampling and Frank–Wolfe optimization—are proposed with theoretical error bounds. Random projections approximate inner products in $O(J)$ per pair for tractability in large $N$. Experimental results on high-dimensional regression and clustering demonstrate significant computational savings per unit accuracy (Campbell et al., 2017).

6. Particle-based and SGMCMC-based Approaches for High Dimensions

Stochastic gradient MCMC (SGMCMC) and particle-based variational methods have become central to scalable Bayesian inference on high-dimensional and deep models. "A Unified Particle-Optimization Framework for Scalable Bayesian Sampling" (Chen et al., 2018) unites SGMCMC and Stein Variational Gradient Descent (SVGD) as solutions to gradient flows in the space of probability measures, governed by energy functionals (e.g., the KL divergence or entropy-regularized potential). This connection clarifies algorithmic design: combining drift, repulsive interactions, and diffusion yields samplers with improved finite-sample performance, as shown by mode coverage in multimodal posteriors and better test performance in Bayesian deep learning (Chen et al., 2018).

In "Scalable Bayesian Learning with posteriors" (Duffield et al., 31 May 2024), the posteriors PyTorch library implements advanced SGMCMC in a tempered setting. The SDE formulation:

$$dz = [D(z) + Q(z)] \frac{1}{N}\nabla\log \pi(z)dt + \mathcal{T}\nabla\cdot[D(z)+Q(z)]dt + \sqrt{2\mathcal{T}D(z)}dw$$

connects Bayesian sampling with stochastic optimization; as $\mathcal{T}\to 0$ the algorithm recovers SGD, while nonzero $\mathcal{T}$ yields unbiased Bayesian posterior sampling. This framework supports principled scalable Bayesian deep ensembles and demonstrates benefits in uncertainty quantification and continual learning with LLMs (Duffield et al., 31 May 2024).

7. Variational and Laplace-Based Approximations with Skewness Correction

In certain latent Gaussian models, the posterior marginals can be highly skewed, e.g., in heavy-tailed likelihoods or binary regression with class imbalance. "Scalable skewed Bayesian inference for latent Gaussian models" (Dutta et al., 26 Feb 2025) introduces a scalable extension to integrated nested Laplace approximations (INLA), wherein some marginals are skew-normal (via transformation $g(\cdot)$), and dependence is retained through the Gaussian copula:

$$p_{SGC}(\tilde f; \mu, Q, s) = p_N(g^{-1}(\tilde f); \mu, Q) \cdot |\mathrm{Jacobian}|$$

Low-rank variational corrections ameliorate mean and covariance errors, and skewness corrections are estimated via variational divergence objectives. Blocking and partial inversions further enhance scalability for large $p$. Empirical results indicate improved accuracy vis-à-vis classical Gaussian INLA or Laplace-approximate posteriors, particularly in asymmetric and rare event regimes (Dutta et al., 26 Feb 2025).

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Summary Table: Key Principles Across Frameworks

Principle/Innovation	Example Reference	Key Feature
Adaptive basis neural surrogates	(Snoek et al., 2015)	Linear in $N$ scaling, cubic only in basis size $D$
Divide-and-conquer + Wasserstein	(Srivastava et al., 2015)	Barycenter aggregation, function-invariant contraction
Product-form factorization	(Atchade, 2015)	Parallel MCMC, optimal contraction via local regressions
Block-diagonal model selection	(Papaspiliopoulos et al., 2016)	Localized inference, exponential-in-block not variable
Hilbert-norm Bayesian coresets	(Campbell et al., 2017)	Inner-product-based coreset, projection scaling
Particle-based/SGMCMC flows	(Chen et al., 2018, Duffield et al., 31 May 2024)	Gradient flows, mini-batch, parallel/ensemble sampling
Variational/laplace w/ skewness	(Dutta et al., 26 Feb 2025)	Skew marginals, copula dependence, variational adjust.

Scalable Bayesian frameworks thus employ a diverse set of algorithmic, algebraic, geometric, and systems-oriented strategies to ensure that uncertainty quantification, model selection, and posterior approximation remain accurate and computationally feasible as models, data, or resources scale. These frameworks have demonstrated substantial impact in hyperparameter optimization, high-dimensional graphical models, fast Bayesian deep learning, large-scale probabilistic programming, high-throughput model selection, optimal experimental design, and complex real-world applications where classical Bayesian approaches are intractable.